Calculus Test for yr 11 PDF

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Name: __TOPIC TESTIntroduction to calculus ( Advanced) Time allowed: 45 minutes Part A: 10 multiple-choice questions (1 0 marks) Part B: 10 free-response questions (4 0 marks) Total: 50 marks Part A10 multiple-choice questions1 mark each: 10 marksCircle the correct answer.1 a =dvdt, where v represen...

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Name:

__________________________________

TOPIC TEST

Introduction to calculus (Advanced)

Part A 10 multiple-choice questions 1 mark each: 10 marks Circle the correct answer.

1

dv , where v represents velocity. dt What does a not stand for?

a=

A

change in v change in t

B instantaneous rate of change of velocity

3

Find the derivative of f (x) = (2x3 + 5)4. A f '(x) = 24x(2x3 + 5)3 B f '(x) = 4x2(2x3 + 5)3 C f '(x) = 24x2(2x3 + 5)3 D f '(x) = 8x(2x3 + 5)3

C f '(v) D acceleration

2

Which one of these functions is differentiable at x = 1? A y = (x + 1)(x – 1) B y=

1 x −1

C y = |x – 1|

x2 + 1, x > 1 D f ( x) =   x +1, x ≤1

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4

5

Which graph below is the gradient function of this graph?

A

B

C

D

Find the difference quotient

f (x + h) − f (x ) for h

8

f (x) = x2 – 4. A –2x + h

B 2x + h

C 2xh

D 2hx + h2 9

6

Differentiate y = 3 x 2 . A

2 dy = 3 dx 3 x

B

2 dy = 3 dx 3 x

C

dy =23 x dx

D

dy 3 = 3 dx 2 x

Differentiate y = x(x2 + 1)4.

Find the value of f '(–4) if f (x) = 2x – x3.

A y' = 8x2(x2 + 1)3

A 56

B 50

B y' = 4x(x2 + 1)3 + (x2 + 1)4

C –50

D –46

C y' = 4x(x2 + 1)3 D y' = (x2 + 1)3 (9x2 + 1)

7

Find the equation of the normal to the function f (x) = 3x2 – 2x + 1 at x = –1. A 6y – x – 49 = 0

B y – 8x – 49 = 0

C 8y – x – 49 = 0

D y – 8x – 14 = 0

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10 A projectile is fired vertically into the air and its height in metres is given by h = 40t – 5t2 + 4, where t is in seconds. Find when its velocity is zero. A 4s

B 5s

C 7.5 s

D 8s

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Part B 10 free-response questions 40 marks Show your working where appropriate.

11 Draw on the same axes the gradient function of this graph.

[2 marks]

12 Differentiate each function from first principles. a f (x) = x2 – 2x ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ b

y=

1 x

______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ [4 marks]

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13 Find the values of x for which each function cannot be differentiated. a

y=

x 2 x −1

______________________________________________________________________________________ b y = x2 + 2x3 + 1, where x ≠ 7 ______________________________________________________________________________________ c y = |x + 2| ______________________________________________________________________________________ [4 marks] 14 Find the derivative of each function after simplifying the function first. a f (x) = (x2 – x)2 ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ b

y=

2x 5 − 6x 4 + 10x 3 + 2x 2 2x 2

______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ [4 marks] 15 Sketch f (x) = x 3 – 8 and f '(x) on the same axes, labelling each graph with its equation and showing all intercepts.

[4 marks]

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16 a

Find the general form of the equation of the tangent to the hyperbola y =

5 at x = 2. x

______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ b There is another tangent to the hyperbola that is parallel to the tangent found in part a . Find the coordinates of the point on the hyperbola at which this tangent exists. ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ [4 marks]

17 f (x) = x3 + 2x2 – x + 1 has a tangent at point P(x, y) that is parallel to the graph of 2x + y = 7. a Find all the possible coordinates of P. ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ b Find the equation of the normal to the graph of y = f (x) at x = –1. ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ [4 marks]

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18 Differentiate each function. a y = (3x2 + 1)(2x – 5) ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ b

g (x ) =

x3 − 1 x +2

______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ c

h (x ) = 2 x 4 − 2 x ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________

d y = (2x2 – 1)2 (7x + 1)3 ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ [8 marks]

19 f (t) = t2 – 12t + 2. a Find the average rate of change between t = 1 and t = 4. ______________________________________________________________________________________ ______________________________________________________________________________________ b Find the instantaneous rate of change at t = 4. ______________________________________________________________________________________ ______________________________________________________________________________________ [2 marks]

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20 A ball is falling from the top of 350 m tower. The height of the ball is represented by h = 350 + 5t – 5t2, where h is in metres and t is time in seconds. a What is the height of the ball after 5 seconds? ______________________________________________________________________________________ ______________________________________________________________________________________ b What is the average speed of the ball between 5 seconds and 10 seconds? ______________________________________________________________________________________ ______________________________________________________________________________________ c What is the speed of the ball in the 10th second? ______________________________________________________________________________________ ______________________________________________________________________________________ d At what speed would the ball hit the ground? Hence, what are the limitations of this model? ______________________________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________ [4 marks]

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